Lecture Notes in Computer Science 3472

266 Verena Wolf
is that an action a is provided by the environment with probability µ i (a). The
occurrence of the action τ in a probabilistic trace simulates that τ is ”performed
by the environment” with probability µ i (τ). Let ɛ denote the empty probabilistic
trace.
The probability of a probabilistic trace is computed in a similar way as the
probability of a trace (see Section 9.3.1) but with a different ”normalization
factor” in each step because we must take into account the probability of an
action in each step. So we have some kind of conditional probability.
We present some helpful definitions now. Let P ∈ FPP , µ ∈D, s, s ′ ∈ SP and
i ∈ N.
• Let v(s,µ)= �
a∈Act Pr a P (s)·µ(a)+Pr τ P (s)+µ(τ) be a normalization factor.
The first summand denotes the probability of performing an observational
action a that is provided with probability µ(a). Pr τ P
(s) is the probability
that P performs an internal action autonomously (independent of µ) and
µ(τ) is the probability that the environment performs τ independently.
• Let Pr silent
P (s, i, s ′ ,µ) denote the probability of reaching s ′ via i τ-transitions
from P ”under the condition µ” whenstartingins:
�
′ 1 if s = s ,
Let
Pr silent
P (s, 0, s ′ ,µ)=
Pr silent
P (s, i +1, s ′ ,µ)=
Pr silent
P (s, s ′ ,µ)= ∞�
i=0
0 otherwise.
⎧
⎪⎨
1 �
v(s,µ) · Pr
ˆs∈SP
τ P
silent
(s, ˆs)· PrP (ˆs, i, s ′ ,µ)
if v(s,µ) > 0,
⎪⎩
0 otherwise.
Pr silent
P (s, i, s ′ ,µ)
denote the probability of reaching s ′ via a sequence of τ-transitions when
starting in s ”under the condition µ”. Note that Pr silent
P (s, s ′ ,µ) is welldefined
because P is divergence-free.
• For a �= τ let
�
Pr a P (s, s′ ,µ)= Pr
ˆs:v(ˆs,µ)>0
silent
P (s, ˆs,µ) · Pr a P (ˆs, s′ ) · µ(a)
v(ˆs,µ) ·
be the probability of a ”weak” a-transition, i.e. s ′ is reached via a (possibly
empty) sequence of τ-transitions followed by one a-transition when starting
in s.
Fora”weak”τ-transition performed ”by the environment” (simulated by µ)
we have
Pr τ P (s, s′ ,µ)=Pr silent (s, s ′ ,µ) ·
µ(τ )
v(s ′ ,µ) .